ITP-UU-07/17
SPIN-07/10
MODELS FOR CONFINEMENT
∗
Gerard ’t Hooft
Institute for Theoretical Physics
Utrecht University
and
Spinoza Institute
Postbox 80.195
3508 TD Utrecht, the Netherlands
e-mail: g.thooft@phys.uu.nl
internet: http://www.phys.uu.nl/~thooft/
Abstract
The dynamical mechanisms causing quarks to be absolutely confined within
the boundaries of a hadronic particle are discussed from various perspectives.
March 19, 2007
∗
Presented at the 50th Anniversary of the Sakata Model, 25-26 November 2006, Nagoya, Japan
1
1.
Introduction: The Sakata Model
According to the model proposed by Sakata[1], there could be a relation between baryonic
particles, then thought to be represented by the proton ( P ), the neutron ( N ) and the
lambda ( Λ ) on the one hand, and the leptons, ν, e− and µ− on the other:
P = hB + νi ,
N = hB + e− i ,
Λ = hB + µ− i .
(1.1)
The distinction between νe and νµ was of course not yet known. B + must have been
thought of as being some boson with lepton number −1 and baryon number 1 (a “leptobaryon”), and clearly it must have been thought to be sufficiently tightly bound to the
leptons to escape detection at that time. What kinds of forces could have been responsible
for that?
The modern versions of our models for baryons and mesons assume them to be composed of quarks. In the 1960s, the arguments used were of the same algebraical nature as
those that led Sakata to his model. Then too, the question of the nature of the binding
force was posed. When it was assumed that the constituents were strongly charged, and
that the binding was caused by the attraction between the charges[2][3], this could only
be part of an answer. In contrast with the electromagnetic case, it seems to be impossible
to “ionize” hadronic particles: individual quarks are never produced. Why should this be
so?
2.
Absolute quark confinement in lattice QCD
One of the first indications that quark confinement really might be absolute, so that
ionization leading to isolated quarks is fundamentally impossible, came from the lattice
formulation of Quantum Chromodynamics[4]. In the lattice theory, the continuum of
space and time is replaced by a lattice of individual points x ,of which the nearest
neighbors, x and x ± a eµ are connected by links [x, x ± a eµ ] , where the index µ is
a Lorentz index, and a is the size of the meshes of the lattice. The quark fields q i (x)
are defined on the points x . The index i here is the color index (the Dirac spin index
is suppressed here). The gauge field Aµ (x) is replaced by a connection operator U (x, µ)
defined on the link [x, x + a eµ ] :
µ Z x+eµ
¶
def
µ
U (x, µ) = exp ig
Aµ dx
.
(2.1)
x
The same link in the opposite direction, by definition, corresponds to the inverse of the
group element U . In the limit of small lattice length a , we consider the links at the
boundary of an elementary plaquette in the 1-2 direction:
¢
¡
2 4
a + ··· .
(2.2)
U1 U2 U−1 U−2 → exp igF12 δx1 ∧ δx2 = I + igF12 a2 − 12 g 2 F12
1
This enables one to approximate the QCD action, − 14 Tr (Fµν Fµν ) by
1 X
Tr (U U U U )µν − C ,
2g 2 x,µ,ν
(2.3)
where we also sum over the orientations of the plaquettes (giving six terms), such that
in Eq. (2.2) the terms linear in Fµν cancel out. The constant C is chosen so as to cancel
Tr (I) but in general it has no effect.
The QCD amplitudes are now described in terms of the functional integrals over the
field variables, with the exponent of this action as integrand. Now, we consider the 1/g 2
expansion in the Euclidean case. It allows us to expand the exponent of the action, so
that integrals of the type
Z
DqDqDU q i (x1 ) U U · · · U q(x2 ) ,
(2.4)
are encountered. By merely applying group symmetry, one derives the general outcome
of such integrals:
Z
X
dU Uip Ujq · · · Ur†k Us†` · · · =
CPerm δsk δj` · · · δrp δsq · · · ,
(2.5)
Perm
where the sum is over various allowed permutations between the upper indices only. The
coefficients CPerm can readily be determined. Local gauge-invariance requires indices to
match pairwise at every lattice point separately. If at the l.h.s. of Eq. (2.5), the indices
cannot be paired up point by point, the integral vanishes. Now, of we deal with a quark
field q i (x1 ) and an antiquark field q j (x2 ) , then the only expressions where the indices
can be paired, are the ones where all U fields occur, on all links that connect x1 with
x2 .
In the functional integral, this implies that, if we have a quark propagator running
from x1 (0) to x1 (t) and an antiquark propagator running from x2 (0) to x2 (t) , the only
non-vanishing contribution to the functional integral comes from configurations where a
sequence of plaquettes contributes in such a way that the plaquettes form a closed surface
with the quark lines at its edges. Since each plaquette gives a suppression factor 1/g 2
(which were assumed to be small in the 1/g 2 expansion), we obtain a total suppression
factor equal to 1/g 2 raised to a power which is equal to the number of plaquettes needed
to produce the surface in between, see Fig. 1. This factor can be written as
(1/g 2 )L t = e−2t L log(1/g) = e−tV (x1 ,x2 ) ,
(2.6)
where L is the distance between the quark lines, and t the interval in the Euclidean
time direction. In Euclidean space, this generates a matrix element of the operator e−tH .
Apparently, all these matrix elements of H contain a potential term V increasing linearly
with the distance L between the quark lines. Thus, one obtains a confining potential.
Note that this argument explaining absolute confinement could be misleading. Indeed,
we could apply the same argument to the lattice version of Maxwell’s theory of electromagnetism. Would one not also conclude that the electromagnetic force gives absolute
2
t
q
q
0
L
Figure 1: Lattice plaquettes contributing to the functional integral for an
amplitude containing a quark antiquark pair at Euclidean time τ = 0 and
τ = t . Solid lines are the quark trajectories. In this picture, all plaquettes
in between the quark lines contribute, but in four-dimensional space-time, the
contributing plaquettes may form any closed surface connecting the two lines.
The surface area is bounded by t L , where L is the average distance between
the quarks.
confinement? That is obviously false. Where does the argument then fail? The answer
to this question is that one would be replacing the Maxwell vector potential Aµ by the
connection matrices U (x, µ) . Maxwell’s action would be replaced by a non-quadratic
action (2.3), and this would allow for new solutions: magnetic monopoles. Thus, the lattice action allows for unphysical particles. For large values of the fine structure constant
α , these could indeed lead to absolute confinement, as is further explained in the next
section. For small values of α , the 1/α expansion as the one used here diverges too
strongly to be reliable.
3.
Absolute quark confinement as a topological phenomenon
We conclude from the above that the lattice approximation shows how absolute quark
confinement may take place, but that the argument is not conclusive. It would be helpful
to see how a similar phenomenon could be described in a strictly continuous theory. An
indication as to what might happen can be obtained from the familiar Higgs theory[5].
Consider first the Abelian Higgs model. The Lagrangian is chosen to be
L(A, ϕ) = − 14 Fµν Fµν − Dµ ϕ∗ Dµ ϕ − V (ϕ) ,
where, as usual,
Fµν − ∂µ Aν − ∂ν Aµ ,
Dµ ϕ = (∂µ + iqAµ )ϕ ;
3
(3.1)
V (ϕ) = 12 (ϕ∗ ϕ − F 2 )2 .
(3.2)
q is the unit of electric charge, and F is the vacuum expectation value of the Higgs field
ϕ . This Lagrangian allows for classical solutions where the Higgs field ϕ makes a full
twist:
ϕ(x, y, z, t) → %(x2 + y 2 )
x + iy
,
|x + iy|
(3.3)
where the continuous function % has to vanish when its argument vanishes, but the Higgs
field quickly approaches the vacuum value % → hϕi0 = F when its argument is large. This
configuration is a vortex, and, assuming that the scalar field must obey some continuity
conditions, one can easily convince oneself that this vortex is stable. The energy of the
vortex can easily be estimated, and is found to be proportional to its length in the z direction.[6]
Now imagine this theory to possess magnetic monopoles.[7] A magnetic monopole with
magnetic charge 2π/q is defined to be an object that is attached to a Dirac string.[8]
The Dirac string is essentially just a gauge artifact; putting such a string along the z
x+iy
axis, a gauge transformation of the form Λ(x, y, z, t) → |x+iy|
will remove the string. We
now notice that this gauge transformation would also replace the ϕ field of a vortex by
a field that can be dissolved in the vacuum, which has ϕ → F everywhere. Apparently,
magnetic monopoles also form end points of our vortex. Indeed, the vortex can be seen to
carry a magnetic flux equalling exactly 2π/q . We end up with a picture where magnetic
monopoles are tied together by vortex lines.
Indeed, in the Higgs model, photons are massive so that all fields are short-range.
Long-range magnetic fields should also not be allowed, but that would clash with the law
~ = 0 . We see that, when monopoles are present,
of magnetic flux conservation: div B
their magnetic flux is carried away by vortices.
Since a vortex carries energy proportional to its length, one finds that in a Higgs theory,
magnetic monopoles are absolutely confined, exactly in the way one expects quarks to be
confined in QCD. This idea was first demonstrated by Nielsen and Olesen.[6]
Quarks, however, do not carry large amounts of magnetic charge. Since QCD is
asymptotically free,[9] one actually expects the electric coupling constant q for quarks
to be small in the small-distance regime. Since the unit of magnetic charge is 2π/q ,
quarks would be very strongly interacting at small distances, if they would indeed be
‘color magnetic’ monopoles.
There is of course a better theory for the confinement of quarks. What we have to do
is replace all magnetic charges and fields by electric charges and vice versa.[10] It is the
‘Higgs field’ that we should assume to be magnetically charged, whereas the quarks carry
only color-electric charges. Such a theory would work if we could show that scalar fields
with pure color-magnetic monopole charges exist in QCD, and that their self interaction
is driven by a potential similar to the one of Eq. (3.2).
The first, we can do. Non-Abelian gauge theories naturally carry field configurations
that have color-magnetic monopole charges. In the case where the gauge group is SU (2) ,
4
N
S
a
b
+
−
Figure 2: a )Confinement of magnetic monopoles in a superconductor. The
Higgs field makes a full rotation if we follow a line around the vortex. b )
Confinement of (color-) electric xharges when magnetic monopoles condense
to produce magnetic superconductivity.
one may assume the presence of some scalar field configuration that transforms as the
adjoint representation I = 1 of the gauge group. If it is not explicitly present, it might
be formed by combining gluon fields, which are also in the adjoint representation. If it is
explicitly present then one can describe magnetic monopoles in perturbation theory and
their properties can be calculated. In QCD, we do not expect scalar fields in the adjoint
representation to be present explicitly, but if we formulate the theory in the Abelian projection gauge,[11] we find artificial singularities that play the role of magnetic monopoles.
It is these singularities that we may assume to condense just like Higgs particles, and then
the electric-magnetic dual of the magnetic vortex confinement mechanism is established.
It is this mechanism by which quarks can be absolutely confined.
A disadvantage of the Abelian projection procedure to describe hadrons in QCD is that
it does not seem to be very accessible to systematic perturbative calculations. One has to
assume that the singularities carrying color-magnetic charge will Bose-condense, but firstprinciple derivations are hard. In the lattice theory, first principle derivations are possible,
but require the 1/g 2 expansion, while g 2 is not really large enough. Furthermore, the
lattice breaks several of the good symmetries of our particle models, such as rotation invariance, Lorentz invariance and the chiral part of the flavor group SU (NF )L ⊗SU (NF )R .
We still want something better.
5
4.
A chain of gluons
We will claim that permanent confinement does not really conflict with the standard
perturbation expansion. First, let us see how far we can push a very intuitive picture.
Confining gluonic forces can be imagined to be structured as follows. Consider a spectator
quark and spectator antiquark. We call a quark spectator quark if pair production of this
quark species in the vacuum can be ignored, such as is the case when we endow these
quarks with a large algebraical quark mass mq .
Imagine that we slowly pull these quarks apart. Will the energy of the gluonic fields
in their environment tend to infinity when the interquark distance goes to infinity? There
is actually no very strong reason why this should not be the case. It happens to be so
that the energy remains finite in perturbative Abelian gauge theories, but in the nonAbelian case the infrared structure of the theory is far too complicated to tell for sure.
Greensite and Thorn[12] conjectured that more and more gluons will be created in the
region between the quarks. They make an ansatz for their wave functions. Let the quarks
be fixed on the positions x0 and xN . We take gluons to be at x1 , x2 , · · · , xN −1 . Their
ansatz wave function is
ψ(~x1 , ~x2 , · · · , ~xN −1 ) = A
~ui = ~xi − ~xi−1 ,
N
Y
i=1
ϕ(~ui ) ,
(4.1)
where ϕ(~u) is a universal wave function whose value follows from the extremal principle.
The Hamiltonian contains P
a kinetic part
T (kin) and a potential part V (Coulomb) . One
P
minimizes the energy E = N T (kin) + N V (Coulomb) , both with respect to the form of
the fnction ϕ and the number of gluons N −1 . Then, one systematically tries to improve
the ansatz.
There will be problems with this program. It is hard to see why N has to tend to
infinity when the separation is large, and why the omission of color non-singlets would
not violate unitarity. Could we do this more systematically? A very important question
in this respect is what to choose as our “Coulomb potential” and why. We claim that
this will have to be a confining potential from the start, but can this be justified and
quantified?
There may well be more systematic ways to do this. The point we wish to stress here is
that the expected infrared behavior can indeed be incorporated in a perturbative scheme
such that better convergence may be expected. What we would advocate is to be called
infrared renormalization.[13]
Compare what is usually done in UV renormalization procedures: we replace the
Lagrangian L by
L(g, A) = Lbare + ∆L − ∆L .
(4.2)
Here Lbare is the bare Lagrangian, which may actually be quite different from the effective
Lagrangian that we expect to see in the lowest order expressions of the amplitudes. We
6
take Lbare + ∆L as out lowest order Lagrangian. We then find that infinities cancel
provided that we choose a counter term −∆L . So the effective Lagrangian Lbare + ∆L is
taken as our lowest order Lagrangian. This is just standard renormalization theory. The
renormalization group is obtained when it is found that ∆L has to be chosen differently
at different scales.
This procedure, adding and subsequently subtracting the same term ∆L to our Lagrangian, can also be done for infrared divergences. We write
def
L(A) = − 14 Fµν Fµν + Lgaugefix + · · · = L0 − ∆L ,
(4.3)
where L0 is expected to be the appropriate Lagrangian for the lower order expressions,
which will require −∆L as a counter term later. Choose
Z
0
1
L (A, x) = − 4 dx0 Fµν (x)G(x − x0 )Fµν (x0 ) + · · · .
(4.4)
Pick the radiation gauge:
Lgaugefix = λ∂i Ai + · · · ,
(4.5)
where λ is a Lagrange multiplier. So,
L0 = − 12 (∂µ Ai )G(∂µ Ai ) + 12 (∂i A0 )G(∂i A0 ) + λ∂i Ai + · · · .
A0 now generates a potential V between charges, which obeys
Z
dx0 G(~x − ~x0 )∂i2 V (~x0 − ~y ) = δ 3 (~x − ~y ) .
(4.6)
(4.7)
We now assume that V is a confining potential, typically:
e2 V (~x) =
−α
+ %r + C st ;
r
r = |~x| ,
(4.8)
where % is the effective string constant, to be determined later by imposing consistency.
In ~k space:
4πα V (~k) = −
4πα
8π%
−
.
~k 2
(~k 2 )2
(4.9)
The required Green function G now follows from solving Eq. (4.7), first in ~k space:
³
´−1
2
~
~
~
G(k) = − k V (k)
=
~k 2
~k 2
+ 2%/α
=1−
2%
,
(4.10)
r = |~x − ~x0 | ,
(4.11)
α~k 2
+ 2%
so that
G(~x − ~x0 ) = δ 3 (~x − ~x0 ) −
8π% −√ 2% r
α
e
;
αr
7
the latter part being, of course, just a Yukawa potential.
This way, we find that one has to try the following ∆L :
Z
8π% −√2%/α r
1
∆L(A) = − 4 d3 ~y Fµν (x)
e
Fµν (x + ~y ) ,
αr
r=
p
~y 2 .
(4.12)
We see that this, innocent-looking, extra term ∆L has to be treated exactly like a
renormalization counter term. If all is well, one will find closed loop diagrams to generate
infrared divergences that require just such a term to be cancelled out.
∆L , as derived in Eq. (4.12) looks fairly local and unobtruding, but it does have
drastic consequences for the infrared behavior of the color electric forces, because, in the
zero momentum limit ( ~k → 0 ), it cancels out completely the naive − 14 F F term of the
gauge theory.
The procedure has not yet been tested. We do not know whether this infrared renormalization would suffice to produce meaningful hadronic wave functions in perturbation
expansion, but it appears to be worth trying.
5.
Renormalized gauge-invariant effective actions
In our final chapter, we consider effective theories all by themselves, in the classical (zeroloop) approximation. The Lagrangians to be produced now should be regarded in the
context of the previous discussion. Again, although the real theory is expected to be a
non-Abelian one, the considerations that follow now would apply to Abelian Lagrangians
as well, so, for simplicity, we restrict ourselves to the Abelian case.
Take a vector potential field Aµ and a neutral scalar ϕ . Let the Lagrangian be
L(A, ϕ) = − 41 Z(ϕ) Fµν Fµν − V (ϕ) + Jµ Aµ .
(5.1)
We omit the kinetic term for ϕ . As one may verified later, a kinetic term would not
play any important role in the solutions that follow, other than making them a bit more
complicated to derive. Jµ is an external current — one could think of the charge density
(ρ, ~0) of a stationary separated quark and antiquark pair. We are looking for a stationary
solution:
∂i Di = ρ(x) ;
Di = Z(ϕ)Ei .
(5.2)
The energy density is
H=
1
2
D2
+ V (ϕ) ,
Z(ϕ)
~ field becomes
and since ϕ has no kinetic term, the energy density of the D
µ
¶
2
1 D
U (D) = min 2
+ V (ϕ) .
ϕ
Z(ϕ)
8
(5.3)
(5.4)
By choosing the functions Z and V (relative to one another), we can obtain any kind of
monotonically increasing function of D .
Now, consider a flux tube of the D -field. The conservation law (5.2) ensures that the
field lines are conserved, so the total flux Q = D Σ (where Σ is the surface area over
which D is spread) equals the total charge of a quark or antiquark at the end point of a
flux tube, and it is fixed. The vortex with given total flux Q will spread over a surface
Σ in such a way that the total energy is minimized. Thus, the energy per unit of length
of this vortex will be
µ
¶
U (D)
string
%
= min(Σ U (D)) = min Q
.
(5.5)
D
D
U (D)/D has a non-trivial minimum if, at low values for D , the energy will increase not
faster than linearly in D , rather than quadratically in ordinary electrodyunamics:
U (D) → %string D .
(5.6)
Which functions Z and V will do the job? This is an exercise in Legendre transformations. From (5.4):
dU = 0 →
1 2
D
2
= −
dV
dV
= Z2
.
d(1/Z)
dZ
(5.7)
I we assume some power law relation between V (ϕ) and Z(ϕ) :
V (Z) ≈ Z α ,
α>0,
(5.8)
then we find
1
D ≈ Z 2 (α+1) ;
U ≈ Zα ;
2α
U ≈ D α+1 ,
(5.9)
So, Eq. (5.6) requires α = 1 , that is, a linear dependence between V and Z . The
proportionality constant can then easily be related to the one in (5.6):
V ≈ 21 %2 Z ;
U ≈ %2 Z ≈ %D .
(5.10)
If we keep α < 3 the the energy density of the D field surrounding a charge decreases
slower than 1/r3 so that the total energy is infinite: such theories give confinement,
though flux lines only develop if α ≤ 1 . The Maxwell case has α = ∞ , that is, Z is
fixed. In general, however, Eqs. (5.8)–(5.10) are too crude to be used in detailed models.
Now that we know which relationship we want between Z and V , we can reconsider
the Lagrangian (5.1), and try to eliminate the field ϕ altogether, which might be possible
def
because there is no kinetic term for ϕ . Writing 14 Fµν Fµν = x , and realizing that, so far,
everything we did was classical, we find
L = f (x) = extrϕ (Zx − V ) ,
9
(5.11)
so that we have to solve ϕ out of
x=
dV
.
dZ
(5.12)
Thus, the Lagrangian will be a function of x , and Eq. (5.12) will force V and Z to be
functions of x as well. So, everything depends on only one parameter. Suppose now that
we wish the flux energy % of a unit of flux (see Eq. (5.6)) to be bounded from below, so
as to obtain vortex formation:
def
U (D) = %(D)D ,
% ≥ %0 .
(5.13)
We then ask what kind of function L(x) can be, given the function %(D) . It is then
convenient, temporarily, to use D as the independent variable. We have
V = %D −
1 2
D
2
Z
1 2
D
∂V
= 2 2 ,
∂Z
Z
;
(5.14)
and by differentiating:
dV = d(%D) −
1 2
D dZ
DdD
+ 2 2
Z
Z
→
d(%D)
1
=
.
DdD
Z
(5.15)
∂(%D)
.
∂D
(5.16)
Combining Eqs. (5.11), (5.12) and (5.14), we find
L(x) =
D2
∂%
− %D = D2
;
Z
∂D
√
2x =
Furthermore, using (5.12), we derive
dL = Zdx + xdZ − dV
→ Z=
dL
.
dx
(5.17)
One then discovers that, insisting that % ≥ %0 and also that %D = U (D) → 0 if
D → 0 , the equation for x(D) has no unambiguous inverse, so we cannot write the
Lagrangian as a single-valued function of x = 21 E 2 . Thus, to describe a theory with
confinement, we must keep the functions V and Z in the Lagrangian1 .
6.
Conclusions
Quantum Chromodynamica is an extremely arrurately defined theory. At short distance
scales, due to asymptotic freedom, the forces become weak, so that perturbative treatment
is possible, and calculating the QCD contributions to high-energy scattering processes has
become routine. Its salient property of permanent quark confinement is usually regarded
as a fundamentally non-perturbative feature of the theory; indeed, usual perturbative
1
During the meeting, Ref[13], was cited, where it was stated (Eq. (8.9) and Fig. 9) that ∂L/∂x = 0
at x = 12 %2 ; this, unfortunately, was an error.
10
methods do not allow us to understand it. But perturbation expansion can be performed
in a modified fashion, such that confinement is assumed to be realized from the start.
There are several ways to perform such unusual expansions; they are less systematic,
and there is no guarantee for convergence better than the usual procedure, but one may
suspect improvement since one uses the right spectrum of physical states from the start.
We think that such approaches deserve further study. In any case, one obtains models
that provide us with a good qualitative understanding of how permanent confinement of
quarks can take place. It is not a very mysterious feature of the theory, but rather a
consequence of the behavior of the effective actions at large distances and relatively weak
field strengths.
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[13] G. ’t Hooft, “Confinement of Quarks”, in Proceedings of the 16th International Conference on Particles and Nuclei (Panic’02), Osaka, Japan, Sept.-Oct., 2002, 3c; ITPUU-02/65, SPIN-02/41.
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